WO2018219052A1 - 一种基于对偶模态方程的确定性声固耦合响应预示方法 - Google Patents

一种基于对偶模态方程的确定性声固耦合响应预示方法 Download PDF

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WO2018219052A1
WO2018219052A1 PCT/CN2018/083242 CN2018083242W WO2018219052A1 WO 2018219052 A1 WO2018219052 A1 WO 2018219052A1 CN 2018083242 W CN2018083242 W CN 2018083242W WO 2018219052 A1 WO2018219052 A1 WO 2018219052A1
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acoustic
subsystem
modal
deterministic
coupling
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费庆国
张鹏
李彦斌
吴邵庆
杨轩
姜东�
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东南大学
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    • G06—COMPUTING; CALCULATING OR COUNTING
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
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  • the invention relates to the technical field of acoustic-coupling response prediction, in particular to a deterministic acoustic-solid coupling response prediction method based on a dual-modal equation.
  • test methods theoretical methods and numerical methods can be used to predict the acoustic-solid coupling response of the system under deterministic load excitation. Among them, the test method can obtain reliable results, but the cost of conducting test analysis is high, and the design cycle is long; the theoretical method is only applicable to simple systems, and it is difficult to solve the problem of predicting the acoustic-solid coupling response of complex systems; the numerical method has good performance for complex systems. Applicability is an effective aid to experimental analysis.
  • the dual-modal equation theory uses a fictitious interface to divide the system into coupled subsystems, and based on the finite element calculation subsystem's modality, rather than the modality of the entire coupled system, therefore, the dual modal equation method is more traditional than the finite element.
  • the method has higher analytical efficiency.
  • the present invention discloses a deterministic acoustic-solid coupling response prediction method based on a dual modal equation, which can effectively improve the deterministic load.
  • the acoustic-solid coupling response of the structure under excitation indicates the efficiency.
  • a deterministic acoustic-solid coupling response prediction method based on dual modal equations comprising the following steps:
  • the acoustic-coupling system in the step (1) is a coupling system between the structure and the acoustic cavity, the structural vibration affects the sound pressure pulsation of the acoustic cavity, and the sound pressure pulsation of the acoustic cavity also affects the structural vibration; the structural subsystem is coupled with the acoustic cavity subsystem.
  • the boundary conditions on the interface are approximated to a free state, and the boundary conditions of the acoustic cavity subsystem at the interface with the structural subsystem are approximated as fixed boundaries.
  • the modal parameters and the mode shapes of the structural subsystem and the acoustic cavity subsystem are calculated based on the finite element method.
  • the coupling parameter between the modalities in the adjacent subsystems in the step (3) is calculated by the following formula:
  • W mn is the coupling parameter between the mth-order displacement mode of the structural subsystem and the nth-order acoustic pressure mode of the acoustic cavity subsystem. Is the mode shape of the mth-order displacement mode of the structural subsystem, For the mode of the nth-order sound pressure mode of the acoustic cavity subsystem, S c is the coupling interface between the structural subsystem and the acoustic cavity subsystem, and s is the spatial position.
  • M where M is the m-th sub-structure displacement modal mass, M n is the n-order sound quality stamper state tune subsystem, ⁇ m and [eta] n are modal damping loss factor m and n are modal, ⁇ For the angular frequency, ⁇ m ( ⁇ ) is the participation factor of the modal m, For the participation factor of modal n, F m ( ⁇ ) and F n ( ⁇ ) are the generalized force loads received on the modal m and the modal n, respectively.
  • step (6) the system dual modal equation is written in the form of a block matrix:
  • the kth modal participation factor in the i-th subsystem can be obtained by inverting the coefficient matrix to the left of equation (5):
  • the superscript "-1" indicates the inverse matrix of the matrix.
  • step (7) the displacement response of the structural subsystem is calculated by:
  • step (2-7) only the mode of the subsystem whose natural frequency is lower than the upper limit of the analysis band of 1.25 times is included.
  • the present invention discloses a deterministic acoustic-solid coupling response prediction method based on a dual-modal equation, which is a deterministic acoustic-solid coupling response prediction method superior to the conventional finite element method, which can effectively improve certainty
  • the acoustic-solid coupling response of the structure under load excitation predicts efficiency, shortens the design cycle, and saves design costs.
  • Figure 1 is a logic flow diagram of the present invention
  • Figure 2 is a schematic view of a stiffened plate/acoustic cavity coupling model
  • Figure 3 is a finite element model of a stiffened panel
  • Figure 4 is the acoustic-solid coupling response at the response point in the subsystem under deterministic concentrated force load excitation (a) stiffened plate, (b) acoustic cavity;
  • Figure 5 is the acoustic-solid coupling response at the response point in the subsystem under deterministic noise load excitation (a) stiffened plate, (b) acoustic cavity.
  • Figure 1 shows a logic flow diagram of a deterministic acoustic-solid coupling response prediction method based on the dual modal equation, which mainly includes the following steps:
  • Step (1) divides the structure and acoustic cavity in the acoustic-coupling system into different subsystems;
  • the acoustic-coupling system is a coupling system between the structure and the acoustic cavity
  • the structural vibration affects the sound pressure fluctuation of the acoustic cavity
  • the sound pressure fluctuation of the acoustic cavity also affects the structural vibration.
  • the boundary condition of the structural subsystem at the coupling interface with the acoustic cavity subsystem is approximated to a free state, and the boundary condition of the acoustic cavity subsystem at the coupling interface with the structural subsystem is approximated as a fixed boundary.
  • Step (2) calculates the modality of the structural subsystem and the acoustic cavity subsystem with the natural frequency lower than 1.25 times the upper limit of the analysis band; the modal parameters and mode shapes of the structural subsystem and the acoustic cavity subsystem are calculated based on the finite element method.
  • Step (3) calculates a coupling parameter between modes in which the natural frequency of the adjacent subsystem is less than 1.25 times the upper limit of the analysis band; the specific formula is calculated by:
  • W mn is the coupling parameter between the mth-order displacement mode of the structural subsystem and the nth-order acoustic pressure mode of the acoustic cavity subsystem.
  • W mn is the coupling parameter between the mth-order displacement mode of the structural subsystem and the nth-order acoustic pressure mode of the acoustic cavity subsystem.
  • Step (4) establishes the dual mode equation of the coupled system:
  • M where M is the m-th sub-structure displacement modal mass, M n is the n-order sound quality stamper state tune subsystem, ⁇ m and [eta] n are modal damping loss factor m and n are modal, ⁇ For the angular frequency, ⁇ m ( ⁇ ) is the participation factor of the modal m, For the participation factor of modal n, F m ( ⁇ ) and F n ( ⁇ ) are the generalized force loads received on the modal m and the modal n, respectively.
  • Step (5) obtains the generalized force load received by the subsystem mode under the action of the deterministic load through the pre-processing; when the structural subsystem is excited by the deterministic surface pressure load P(s, ⁇ ), the mode m
  • the generalized force load received is given by:
  • Step (6) calculates a dual mode equation and obtains a participation factor for all modes
  • the kth modal participation factor in the i-th subsystem can be obtained by inverting the coefficient matrix to the left of equation (5):
  • the superscript "-1" indicates the inverse matrix of the matrix.
  • Step (7) calculates the deterministic acoustic-solid coupling response of the system by modal superposition; specifically calculates the displacement response of the structural subsystem by the following formula:
  • the boundary conditions of the stiffened plate are: simply supported on four sides, the finite element model of the stiffened plate is shown in Figure 3; the parameters of the panel of the stiffened plate are given in Table 1, the material parameters of the rib and the material parameters of the panel Similarly, the dimension parallel to the x-axis rib is 1 m ⁇ 0.03 m ⁇ 0.005 m, the pitch is 1 / 6 m, and the dimension of the rib parallel to the y-axis is 1 m ⁇ 0.02 m ⁇ 0.005 m, and the pitch is 1 / 6 m.
  • the boundary conditions of the acoustic cavity are: except for the face coupled with the stiffened plate, the remaining faces are fixed boundaries; the parameters of the acoustic cavity are given in Table 2.
  • a unit deterministic noise load is applied to the outer surface of the stiffened panel, and after the above steps, the acceleration response spectrum at the response point of the coordinate plate (0.3 m, 0.1 m) on the stiffened panel is obtained, and the coordinates in the acoustic cavity are (
  • the sound pressure response spectrum at the response point of 0.3m, 0.1m, 0m) is shown in Fig. 5(a) and Fig. 5(b), respectively.
  • the final result of the embodiment shows that the method proposed by the invention can effectively solve the problem of predicting the acoustic-solid coupling response of the complex system under the deterministic load excitation, and improve the efficiency of the analysis.

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Abstract

一种基于对偶模态方程的确定性声固耦合响应预示方法，包括如下步骤：(1)将声固耦合系统中的结构和声腔划分成不同的子系统；(2)计算结构子系统和声腔子系统的模态；(3)计算相邻子系统中模态间的耦合参数；(4)建立耦合系统的对偶模态方程；(5)通过前置处理，获得确定性载荷作用下，子系统模态上受到的广义力载荷；(6)计算对偶模态方程，获得所有模态的参与因子；(7)通过模态叠加，计算系统确定性声固耦合响应。上述确定性声固耦合响应预示方法，是一种基于对偶模态方程的确定性声固耦合响应预示方法，该方法把系统划分成连续耦合的子系统，并用有限频带内的子系统模态描述系统的确定性振动，该方法的分析效率高于传统有限元法。

Description

一种基于对偶模态方程的确定性声固耦合响应预示方法 技术领域
本发明涉及声固耦合响应预示技术领域，具体涉及一种基于对偶模态方程的确定性声固耦合响应预示方法。
背景技术
航天器在任务周期内面临严峻的机械振动、噪声等环境，这可能造成结构失效或精密仪器、仪表失灵。因此，在航天器的设计过程中，需考虑机械振动和噪声的影响。可采用试验方法、理论方法和数值方法预示系统在确定性载荷激励下的声固耦合响应。其中，试验方法能得到可靠的结果，但开展试验分析的成本较高，设计周期长；理论方法只适用于简单系统，难以解决复杂系统声固耦合响应预示问题；数值方法对复杂系统有良好的适用性，是试验分析的有效辅助手段。对偶模态方程理论用虚构的界面将系统划分成耦合的子系统，并基于有限元计算子系统的模态，而不是整个耦合系统的模态，因此，对偶模态方程方法比传统的有限元法具有更高的分析效率。
现有对偶模态方程理论中存在模态截断问题，即需截取有限频率范围内的子系统模态参与响应预示，所选模态过少会引起误差，所选模态过多会造成计算资源浪费。因此，需要有一个准则界定模态截断的频率范围，以合理地基于对偶模态方程预示系统在确定性载荷激励下的声固耦合响应。
发明内容
发明目的：针对现有的一种对偶模态方程方法在应用中存在的问题，本发明公开了一种基于对偶模态方程的确定性声固耦合响应预示方法，该方法可有效提高确定性载荷激励下结构的声固耦合响应预示效率。
技术方案：为实现上述目的，本发明采用的技术方案为：
一种基于对偶模态方程的确定性声固耦合响应预示方法，包括以下步骤：
(1)将声固耦合系统中的结构和声腔划分成不同的子系统；
(2)计算结构子系统和声腔子系统的模态；
(3)计算相邻子系统中模态间的耦合参数；
(4)建立耦合系统的对偶模态方程；
(5)通过前置处理，获得确定性载荷作用下，子系统模态上受到的广义力载荷；
(6)计算对偶模态方程，获得所有模态的参与因子；
(7)通过模态叠加，计算系统确定性声固耦合响应。
进一步地，所述步骤(1)中的声固耦合系统为结构与声腔耦合系统，结构振动会影响声腔声压脉动，声腔声压脉动也会影响结构振动；结构子系统在与声腔子系统耦合界面上的边界条件被近似为自由状态，声腔子系统在与结构子系统耦合界面上的边界条件被近似为固定边界。
进一步地，所述步骤(2)中基于有限元法计算了结构子系统和声腔子系统的模态参数和模态振型。
进一步地，所述步骤(3)中相邻子系统中模态间的耦合参数由下式计算得到：
Figure PCTCN2018083242-appb-000001
其中W mn为结构子系统第m阶位移模态与声腔子系统第n阶声压模态之间的耦合参数，
Figure PCTCN2018083242-appb-000002
为结构子系统第m阶位移模态的振型，
Figure PCTCN2018083242-appb-000003
为声腔子系统第n阶声压模态的振型，S c为结构子系统与声腔子系统之间的耦合界面，s为空间位置。
进一步地，所述步骤(4)中建立的耦合系统的对偶模态方程为：
Figure PCTCN2018083242-appb-000004
其中M m为结构子系统第m阶位移模态质量，M n为声腔子系统第n阶声压模态质量，η m和η n分别为模态m和模态n的阻尼损耗系数，ω为角频率，φ m(ω)为模态m的参与因子，
Figure PCTCN2018083242-appb-000005
Figure PCTCN2018083242-appb-000006
为模态n的参与因子，F m(ω)和F n(ω)分别为模态m和模态n上受到的广义力载荷。
进一步地，所述步骤(5)中，结构子系统受到确定性的面压载荷P(s,ω)激励时，模态m上受到的广义力载荷由下式给出：
Figure PCTCN2018083242-appb-000007
其中S p为面压载荷作用面。当结构子系统受到确定性集中力载荷F 0(ω)激励时，模态m上受到的广义力载荷由下式给出：
Figure PCTCN2018083242-appb-000008
其中s 0为集中力载荷F 0(ω)的作用位置。
进一步地，所述步骤(6)中将系统对偶模态方程写成了分块矩阵的形式：
Figure PCTCN2018083242-appb-000009
其中上标“T”表示矩阵的转置，
Figure PCTCN2018083242-appb-000010
Figure PCTCN2018083242-appb-000011
W(m,n)＝W mn(8)
Figure PCTCN2018083242-appb-000012
其中diag()表示对角矩阵，括号内为对角矩阵元素。基于式(5)求得各子系统内的模态参与因子：
Figure PCTCN2018083242-appb-000013
其中H ij为传递函数矩阵(i＝1,2；j＝1,2)，矩阵元素H ij(k,l)的含义为：当第j个子系统中第l阶模态上作用单位广义力时，第i个子系统中第k阶模态参与因子。传递函数矩阵可对式(5)左边的系数矩阵求逆获得：
Figure PCTCN2018083242-appb-000014
上标“-1”表示矩阵的逆矩阵。
进一步地，所述步骤(7)中通过下式计算结构子系统的位移响应：
Figure PCTCN2018083242-appb-000015
其中
Figure PCTCN2018083242-appb-000016
通过下式计算声腔子系统的声压响应：
Figure PCTCN2018083242-appb-000017
其中
Figure PCTCN2018083242-appb-000018
进一步地，所述步骤(2～7)中，仅包含固有频率低于1.25倍分析频带上限的子系统的模态。
有益效果：本发明公开了一种基于对偶模态方程的确定性声固耦合响应预示方法，是一种优于传统有限元法的确定性声固耦合响应预示方法，该方法可有效提高确定性载荷激励下结构的声固耦合响应预示效率，缩短设计周期，节约设计成本。
附图说明
图1是本发明的逻辑流程框图；
图2是一个加筋板/声腔耦合模型的示意图；
图3是加筋板的有限元模型；
图4是确定性集中力载荷激励下子系统内响应点处声固耦合响应(a)加筋板，(b)声腔；
图5是确定性噪声载荷激励下子系统内响应点处声固耦合响应(a)加筋板，(b)声腔。
具体实施方式
下面结合附图对本发明作更进一步的说明。
如图1所示为一种基于对偶模态方程的确定性声固耦合响应预示方法逻辑流程框图，主要包括以下步骤：
步骤(1)将声固耦合系统中的结构和声腔划分成不同的子系统；声固耦合系统为结构与声腔耦合系统，结构振动会影响声腔声压脉动，声腔声压脉动也会影响结构振动；结构子系统在与声腔子系统耦合界面上的边界条件被近似为自由状态，声腔子系统在与结构子系统耦合界面上的边界条件被近似为固定边界。
步骤(2)计算结构子系统和声腔子系统中固有频率低于1.25倍分析频带上限的模态；具体基于有限元法计算了结构子系统和声腔子系统的模态参数和模态振型。
步骤(3)计算相邻子系统中固有频率低于1.25倍分析频带上限的模态间的耦合参数；具体由下式计算得到：
Figure PCTCN2018083242-appb-000019
其中W mn为结构子系统第m阶位移模态与声腔子系统第n阶声压模态之间的耦合参数，
Figure PCTCN2018083242-appb-000020
为结构子系统第m阶位移模态的振型，
Figure PCTCN2018083242-appb-000021
(s)为声腔子系统第n阶声压模态的振型，S c为结构子系统与声腔子系统之间的耦合界面，s为空间位置。
步骤(4)建立耦合系统的对偶模态方程：
Figure PCTCN2018083242-appb-000022
其中M m为结构子系统第m阶位移模态质量，M n为声腔子系统第n阶声压模态质量，η m和η n分别为模态m和模态n的阻尼损耗系数，ω为角频率，φ m(ω)为模态m的参与因子，
Figure PCTCN2018083242-appb-000023
Figure PCTCN2018083242-appb-000024
为模态n的参与因子，F m(ω)和F n(ω)分别为模态m和模态n上受 到的广义力载荷。
步骤(5)通过前置处理，获得确定性载荷作用下，子系统模态上受到的广义力载荷；结构子系统受到确定性的面压载荷P(s,ω)激励时，模态m上受到的广义力载荷由下式给出：
Figure PCTCN2018083242-appb-000025
其中S p为面压载荷作用面。当结构子系统受到确定性集中力载荷F 0(ω)激励时，模态m上受到的广义力载荷由下式给出：
Figure PCTCN2018083242-appb-000026
其中s 0为集中力载荷F 0(ω)的作用位置。
步骤(6)计算对偶模态方程，获得所有模态的参与因子；
(6.1)将系统对偶模态方程写成了分块矩阵的形式：
Figure PCTCN2018083242-appb-000027
其中上标“T”表示矩阵的转置，
Figure PCTCN2018083242-appb-000028
Figure PCTCN2018083242-appb-000029
W(m,n)＝W mn              (8)
Figure PCTCN2018083242-appb-000030
其中diag()表示对角矩阵，括号内为对角矩阵元素。
(6.2)基于式(5)求得各子系统内的模态参与因子：
Figure PCTCN2018083242-appb-000031
其中H ij为传递函数矩阵(i＝1,2；j＝1,2)，矩阵元素H ij(k,l)的含义为：当第j个子系统中第l阶模态上作用单位广义力时，第i个子系统中第k阶模态参与因子。传递函数矩阵可对式(5)左边的系数矩阵求逆获得：
Figure PCTCN2018083242-appb-000032
上标“-1”表示矩阵的逆矩阵。
步骤(7)通过模态叠加，计算系统确定性声固耦合响应；具体通过下式计算结构子系统的位移响应：
Figure PCTCN2018083242-appb-000033
其中
Figure PCTCN2018083242-appb-000034
通过下式计算声腔子系统的声压响应：
Figure PCTCN2018083242-appb-000035
其中
Figure PCTCN2018083242-appb-000036
实施例
以一个加筋板/声腔耦合模型为例，如图2所示。加筋板的边界条件为：四条边上简支，加筋板的有限元模型如图3所示；加筋板的面板的参数由表1给出，筋条的材料参数与面板的材料参数相同，平行于x轴向筋条的尺寸为1m×0.03m×0.005m，间距1/6m，平行于y轴向筋条的尺寸为1m×0.02m×0.005m，间距1/6m。声腔的边界条件为：除与加筋板耦合的面，其余各面为固定边界；声腔的参数由表2给出。
在加筋板面板上坐标为(0.2m,0.15m)的点处施加垂直于板面的单位确定性集中力载荷，经过上述各步骤，得到加筋板面板上坐标为(0.3m,0.1m)的响应点处的加速度响应频谱，以及声腔中坐标为(0.3m,0.1m,0m)的响应点处的声压响应频谱，分别如图4(a)和图4(b)所示。
在加筋板面板的外表面施加单位确定性噪声载荷，经过上述各步骤，得到加筋板面板上坐标为(0.3m,0.1m)的响应点处的加速度响应频谱，以及声腔中坐标为(0.3m,0.1m,0m)的响应点处的声压响应频谱，分别如图5(a)和图5(b)所示。
图4和图5中的参考值由有限元直接法计算得到。在对偶模态方程方法分析过程中，选取了2.5kHz以内的加筋板模态和声腔模态参与响应预示。图4和图5中结果显示，基于对偶模态方程能准确预示确定性集中力载荷及确定性噪声载荷激励下系统的声固耦合响应。
本实施例最终取得的效果说明，本发明所提出的方法能有效地解决确定性载荷激励下复杂系统的声固耦合响应预示问题，提高分析的效率。
以上所述仅是本发明的优选实施方式，应当指出：对于本技术领域的普通技术人员来说，在不脱离本发明原理的前提下，还可以做出若干改进和润饰，这些改进和润饰也应视为本发明的保护范围。
表1 加筋板的面板的参数取值
Figure PCTCN2018083242-appb-000037
表2 声腔的参数取值
Figure PCTCN2018083242-appb-000038

Claims (9)

  1. 一种基于对偶模态方程的确定性声固耦合响应预示方法，其特征在于：包括以下步骤：
    (1)将声固耦合系统中的结构和声腔划分成不同的子系统；
    (2)计算结构子系统和声腔子系统的模态；
    (3)计算相邻子系统中模态间的耦合参数；
    (4)建立耦合系统的对偶模态方程；
    (5)通过前置处理，获得确定性载荷作用下，子系统模态上受到的广义力载荷；
    (6)计算对偶模态方程，获得所有模态的参与因子；
    (7)通过模态叠加，计算系统确定性声固耦合响应。
  2. 根据权利要求1所述的基于对偶模态方程的确定性声固耦合响应预示方法，其特征在于：所述步骤(1)中的声固耦合系统为结构与声腔耦合系统，结构振动会影响声腔声压脉动，声腔声压脉动也会影响结构振动；结构子系统在与声腔子系统耦合界面上的边界条件被近似为自由状态，声腔子系统在与结构子系统耦合界面上的边界条件被近似为固定边界。
  3. 根据权利要求1所述的基于对偶模态方程的确定性声固耦合响应预示方法，其特征在于：所述步骤(2)中基于有限元法计算了结构子系统和声腔子系统的模态参数和模态振型。
  4. 根据权利要求1所述的基于对偶模态方程的确定性声固耦合响应预示方法，其特征在于：所述步骤(3)中相邻子系统中模态间的耦合参数由下式计算得到：
    Figure PCTCN2018083242-appb-100001
    其中W mn为结构子系统第m阶位移模态与声腔子系统第n阶声压模态之间的耦合参数，
    Figure PCTCN2018083242-appb-100002
    为结构子系统第m阶位移模态的振型，
    Figure PCTCN2018083242-appb-100003
    为声腔子系统第n阶声压模态的振型，S c为结构子系统与声腔子系统之间的耦合界面，s为空间位置。
  5. 根据权利要求1所述的基于对偶模态方程的确定性声固耦合响应预示方法，其特征在于：所述步骤(4)中建立的耦合系统的对偶模态方程为：
    Figure PCTCN2018083242-appb-100004
    其中M m为结构子系统第m阶位移模态质量，M n为声腔子系统第n阶声压模态质量，η m和η n分别为模态m和模态n的阻尼损耗系数，ω为角频率，φ m(ω)为模态m的参与因子，
    Figure PCTCN2018083242-appb-100005
    为模态n的参与因子，F m(ω)和F n(ω)分别为模态m和模态n上受到的广义力载荷。
  6. 根据权利要求1所述的基于对偶模态方程的确定性声固耦合响应预示方法，其特征在于：所述步骤(5)中，结构子系统受到确定性的面压载荷P(s,ω)激励时，模态m上受到的广义力载荷由下式给出：
    Figure PCTCN2018083242-appb-100006
    其中S p为面压载荷作用面。当结构子系统受到确定性集中力载荷F 0(ω)激励时，模态m上受到的广义力载荷由下式给出：
    Figure PCTCN2018083242-appb-100007
    其中s 0为集中力载荷F 0(ω)的作用位置。
  7. 根据权利要求1所述的基于对偶模态方程的确定性声固耦合响应预示方法，其特征在于：所述步骤(6)中将系统对偶模态方程写成了分块矩阵的形式：
    Figure PCTCN2018083242-appb-100008
    其中上标“T”表示矩阵的转置，
    Figure PCTCN2018083242-appb-100009
    Figure PCTCN2018083242-appb-100010
    W(m,n)＝W mn  (8)
    Figure PCTCN2018083242-appb-100011
    其中diag()表示对角矩阵，括号内为对角矩阵元素。基于式(5)求得各子系统内的模态参与因子：
    Figure PCTCN2018083242-appb-100012
    其中H ij为传递函数矩阵(i＝1,2；j＝1,2)，矩阵元素H ij(k,l)的含义为：当第j个子系统中第l阶模态上作用单位广义力时，第i个子系统中第k阶模态参与因子。传递函数矩阵可对式(5)左边的系数矩阵求逆获得：
    Figure PCTCN2018083242-appb-100013
     上标“-1”表示矩阵的逆矩阵。
  8. 根据权利要求1所述的基于对偶模态方程的确定性声固耦合响应预示方法，其特征在于：所述步骤(7)中通过下式计算结构子系统的位移响应：
    Figure PCTCN2018083242-appb-100014
    其中
    Figure PCTCN2018083242-appb-100015
    通过下式计算声腔子系统的声压响应：
    Figure PCTCN2018083242-appb-100016
    其中
    Figure PCTCN2018083242-appb-100017
  9. 根据权利要求1所述的基于对偶模态方程的确定性声固耦合响应预示方法，其特征在于：所述步骤(2～7)中，仅包含固有频率低于1.25倍分析频带上限的子系统的模态。
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